

Tangent to a circle at a point
Geometry construction using a compass and straightedge
This shows how to construct the
tangent to a circle at a given point on the circle with compass and straightedge or ruler. It works by using the fact that a tangent to a circle is
perpendicular
to the
radius
at the point of contact. It first creates a radius of the circle, then
constructs a perpendicular
to the radius at the given point.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof
The image below is the final drawing above.

Argument 
Reason 
1 
Line segment OR is a radius of the circle O. 
It is a line from the center to the given point P on the circle. 
2 
SP is perpendicular to OR 
By construction, SP is the perpendicular to OR at P.
See Constructing a perpendicular to a line at a point for method and proof.

3 
SP is the tangent to O at the point P 
The tangent line is at right angles to the radius at the point of contact.
See Tangent line definition. 
 Q.E.D
Try it yourself
Click here for a printable tangents problem worksheet with some problems to try.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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